1 Evolution & Economics No. 5 Forest fire p = 0.53 p = 0.58

2 Best Response Dynamics A finite population playing a (symmetric) game G. Each individual is randomly matched and plays a pure strategy of G. At each point in (discrete) time each individual (or alternatively each with probability) chooses the best response to the mixed strategy played by the population.

3 AB A3, 30, x Bx, 0x, x Payoff Dominant Equilibrium AB A3, 30, x Bx, 0x, x 3 > x > 0 Nash Equilibria Risk Dominant Equilibrium

4 Payoff Dominant Equilibrium Risk Dominant Equilibrium AB A3, 30, x Bx, 0x, x A Population of n individuals playing either A or B. A B All playing A All playing B 0 1 k/n k playing B

5 Payoff Dominant Equilibrium Risk Dominant Equilibrium AB A3, 30, x Bx, 0x, x AB A3, 30, 2 B2, 02, α α A B

6 Payoff Dominant Equilibrium Risk Dominant Equilibrium AB A3, 30, x Bx, 0x, x AB A3, 30, 2 B2, 02, A B In the Best Response Dynamics: The Basin of Attraction of the Risk Dominant Equilibrium is larger than that of the Pareto Dominant Equilibrium

7 A B

8 A B A deterministic dynamics can be described as a transition function

9 The replicator Dynamics (in which all learn – revise their strategy) is: If each individual has probability λ of learning: A B

10 If each individual has probability of learning: If each individual has probability λ of learning: A B A state of the population becomes a distribution of states

11 The prob. of learning may be dropped. A B If each individual (when learning) has a small probability ε of learning the ‘wrong’ strategy: B (B) B (A) Each state has a positive probability of becoming any other state.

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14 A B AB A3, 30, 2 B2, 02, 2 ε -small, mistakes are rare The process converges to B (A) and will stay there a long time. In the long-run mistakes occur (with prob. 1 ) and the process will be swept away from B.

15 A B If more individuals made mistakes, the process moves to the basin of attraction of A, from there to A, and will remain there a long time. If few individuals make mistakes, the process will remain in the basin of attraction of B, and will return to B.

16 A B What proportion of the time is spent in A ?? The process remains a long time in A and in B. To leave the basin of attraction of B, a t least 2/3 of the population has to err. This happens with prob.: To leave the basin of attraction of A, at least 1/3 of the population has to err. This happens with prob. :

17 A B It is a rare move from B to the basin of attraction of A. The reverse is much more frequent !!!

18 A B In the very long run the process will spend infinitely more time in B than in A. In the very long run the process will spend 0 time in A.

19 The Best Response Dynamics – with vanishing noise, chooses the Risk Dominant Equilibrium. The equilibrium with a larger Basin of attraction. AB A3, 30, 2 B2, 02, 2 AB A3, 30, 2 B2, 02, 2

20 Kandori, Michihiro & Mailath, George J & Rob, Rafael, "Learning, Mutation, and Long Run Equilibria in Games," Econometrica, Econometric Society, vol. 61(1), pages H Peyton, "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages M.I. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems, Springer New York, 1984 Based on: